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5.1 — Operator precedence and associativity

Chapter introduction

This chapter builds on top of the concepts from lesson 1.8 -- Introduction to literals and operators. A quick review follows:

In mathematics, an operation is a mathematical calculation involving zero or more input values (called operands) that produces a new value (called an output value). The specific operation to be performed is denoted by a construct (typically a symbol or pair of symbols) called an operator.

For example, as children we all learn that 2 + 3 equals 5. In this case, the literals 2 and 3 are the operands, and the symbol + is the operator that tells us to apply mathematical addition on the operands to produce the new value 5.

In this chapter, we’ll discuss topics related to operators, and explore many of the common operators that C++ supports.

Operator precedence

Now, let’s consider a more complicated expression, such as 4 + 2 * 3. In order to evaluate this expression, we must understand both what the operators do, and the correct order to apply them. The order in which operators are evaluated in a compound expression is determined by an operator’s precedence. Using normal mathematical precedence rules (which state that multiplication is resolved before addition), we know that the above expression should evaluate as 4 + (2 * 3) to produce the value 10.

In C++, when the compiler encounters an expression, it must similarly analyze the expression and determine how it should be evaluated. To assist with this, all operators are assigned a level of precedence. Operators with the highest level of precedence are evaluated first.

You can see in the table below that multiplication and division (precedence level 5) have more precedence than addition and subtraction (precedence level 6). Thus, 4 + 2 * 3 evaluates as 4 + (2 * 3) because multiplication has a higher level of precedence than addition.

Operator associativity

What happens if two operators in the same expression have the same precedence level? For example, in the expression 3 * 4 / 2, the multiplication and division operators are both precedence level 5. In this case, the compiler can’t rely upon precedence alone to determine how to evaluate the result.

If two operators with the same precedence level are adjacent to each other in an expression, the operator’s associativity tells the compiler whether to evaluate the operators from left to right or from right to left. The operators in precedence level 5 have an associativity of left to right, so the expression is resolved from left to right: (3 * 4) / 2 = 6.

Table of operators

The below table is primarily meant to be a reference chart that you can refer back to in the future to resolve any precedence or associativity questions you have.

Notes:

  • Precedence level 1 is the highest precedence level, and level 17 is the lowest. Operators with a higher precedence level get evaluated first.
  • L->R means left to right associativity.
  • R->L means right to left associativity.
Prec/Ass Operator Description Pattern
1 None ::
::
Global scope (unary)
Namespace scope (binary)
::name
class_name::member_name
2 L->R ()
()
()
{}
type()
type{}
[]
.
->
++
––
typeid
const_cast
dynamic_cast
reinterpret_cast
static_cast
sizeof…
noexcept
alignof
Parentheses
Function call
Initialization
Uniform initialization (C++11)
Functional cast
Functional cast (C++11)
Array subscript
Member access from object
Member access from object ptr
Post-increment
Post-decrement
Run-time type information
Cast away const
Run-time type-checked cast
Cast one type to another
Compile-time type-checked cast
Get parameter pack size
Compile-time exception check
Get type alignment
(expression)
function_name(parameters)
type name(expression)
type name{expression}
new_type(expression)
new_type{expression}
pointer[expression]
object.member_name
object_pointer->member_name
lvalue++
lvalue––
typeid(type) or typeid(expression)
const_cast<type>(expression)
dynamic_cast<type>(expression)
reinterpret_cast<type>(expression)
static_cast<type>(expression)
sizeof…(expression)
noexcept(expression)
alignof(Type)
3 R->L +
-
++
––
!
~
(type)
sizeof
co_await
&
*
new
new[]
delete
delete[]
Unary plus
Unary minus
Pre-increment
Pre-decrement
Logical NOT
Bitwise NOT
C-style cast
Size in bytes
Await asynchronous call
Address of
Dereference
Dynamic memory allocation
Dynamic array allocation
Dynamic memory deletion
Dynamic array deletion
+expression
-expression
++lvalue
––lvalue
!expression
~expression
(new_type)expression
sizeof(type) or sizeof(expression)
co_await expression
&lvalue
*expression
new type
new type[expression]
delete pointer
delete[] pointer
4 L->R ->*
.*
Member pointer selector
Member object selector
object_pointer->*pointer_to_member
object.*pointer_to_member
5 L->R *
/
%
Multiplication
Division
Modulus
expression * expression
expression / expression
expression % expression
6 L->R +
-
Addition
Subtraction
expression + expression
expression - expression
7 L->R <<
>>
Bitwise shift left
Bitwise shift right
expression << expression
expression >> expression
8 L->R <=> Three-way comparison expression <=> expression
9 L->R <
<=
>
>=
Comparison less than
Comparison less than or equals
Comparison greater than
Comparison greater than or equals
expression < expression
expression <= expression
expression > expression
expression >= expression
10 L->R ==
!=
Equality
Inequality
expression == expression
expression != expression
11 L->R & Bitwise AND expression & expression
12 L->R ^ Bitwise XOR expression ^ expression
13 L->R | Bitwise OR expression | expression
14 L->R && Logical AND expression && expression
15 L->R || Logical OR expression || expression
16 R->L throw
co_yield
?:
=
*=
/=
%=
+=
-=
<<=
>>=
&=
|=
^=
Throw expression
Yield expression
Conditional
Assignment
Multiplication assignment
Division assignment
Modulus assignment
Addition assignment
Subtraction assignment
Bitwise shift left assignment
Bitwise shift right assignment
Bitwise AND assignment
Bitwise OR assignment
Bitwise XOR assignment
throw expression
co_yield expression
expression ? expression : expression
lvalue = expression
lvalue *= expression
lvalue /= expression
lvalue %= expression
lvalue += expression
lvalue -= expression
lvalue <<= expression
lvalue >>= expression
lvalue &= expression
lvalue |= expression
lvalue ^= expression
17 L->R , Comma operator expression, expression

You should already recognize a few of these operators, such as +, -, *, /, (), and sizeof. However, unless you have experience with another programming language, the majority of the operators in this table will probably be incomprehensible to you right now. That’s expected at this point. We’ll cover many of them in this chapter, and the rest will be introduced as there is a need for them.

Q: Where's the exponent operator?

C++ doesn’t include an operator to do exponentiation (operator^ has a different function in C++). We discuss exponentiation more in lesson 5.3 -- Modulus and Exponentiation.

Parenthesization

In normal arithmetic, you learned that you can use parenthesis to change the order of application of operations. For example, we know that 4 + 2 * 3 evaluates as 4 + (2 * 3), but if you want it to evaluate as (4 + 2) * 3 instead, you can explicitly parenthesize the expression to make it evaluate the way you want. This works in C++ because parenthesis have one of the highest precedence levels, so parenthesis generally evaluate before whatever is inside them.

Now consider an expression like x && y || z. Does this evaluate as (x && y) || z or x && (y || z)? You could look up in the table and see that && takes precedence over ||. But there are so many operators and precedence levels that it’s hard to remember them all. In order to reduce mistakes and make your code easier to understand without referencing a precedence table, it’s a good idea to parenthesize any non-trivial compound expression, so it’s clear what your intent is.

Best practice

Use parenthesis to make it clear how an expression should evaluate, even if they are technically unnecessary.

Quiz time

Question #1

You know from everyday mathematics that expressions inside of parentheses get evaluated first. For example, in the expression (2 + 3) * 4, the (2 + 3) part is evaluated first.

For this exercise, you are given a set of expressions that have no parentheses. Using the operator precedence and associativity rules in the table above, add parentheses to each expression to make it clear how the compiler will evaluate the expression.

Show Hint

Sample problem: x = 2 + 3 % 4

Binary operator % has higher precedence than operator + or operator =, so it gets evaluated first:

x = 2 + (3 % 4)

Binary operator + has a higher precedence than operator =, so it gets evaluated next:

Final answer: x = (2 + (3 % 4))

We now no longer need the table above to understand how this expression will evaluate.

a) x = 3 + 4 + 5;

Show Solution

b) x = y = z;

Show Solution

c) z *= ++y + 5;

Show Solution

d) a || b && c || d;

Show Solution


5.2 -- Arithmetic operators
Index
4.x -- Chapter 4 summary and quiz

155 comments to 5.1 — Operator precedence and associativity

  • Martin

    Found a typo in the reference table, precedence level 8:
    expresstion <=> expression

  • Shtimpo

    nascardriver or Alex, the operator precedence table at https://en.cppreference.com/w/cpp/language/operator_precedence
    looks a bit different than the one you have here.
    Not sure which one is more correct/updated, only pointing it out.
    (I was trying to find out which one is the ternary operator, but ended up finding more than that)

    • nascardriver

      I moved `throw` into precedence level 16 and added the missing operators. Is there anything more that you noticed in this table?
      Thanks!

      • Shtimpo

        Nope, nothing else other than the fact that yours looks more organised as it shows the operators more clearly on separate rows. Thanks

  • keshav

    Who are you why this non-sence?

  • fg

  • sami (insertion and extraction operators)

    Why do we have "Bitwise shift left/right" operators but not "insertion and extraction operator" in the chart above?

    • nascardriver

      It's the same operator. For custom types, like streams, operators can be changed to do whatever the developer wants. For streams, the behavior is usually changed to input and output. We cover this later when we talk about operator overloading.

  • Dexter

    z *= ++y + 5;

    ++y is pre-increment. the pattern is ++lvalue. It has lower precedence than unary operator +.
    y++ is post-increment. the pattern is lvalue++. It has higher precedence than unary operator +.
    *= the lowest precedence.
    Therefore:

    z *= (++(y + 5));

    Please correct me if I'm wrong. Thank you very much.

    • nascardriver

      Lower precedences get executed first. ++y get executed before `+ 5`. The correct order is

    • Vy

      I think + in this case is in the precedence level 6 since the unary operator is for identifying if a number is positive or negative. So the order is:
      z *= ((++y) + 5)

  • what about ambiguity in SLR,LR and operator precedence

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